Showing papers on "Covariance mapping published in 2014"

Tests for Covariance Matrices in High Dimension with Less Sample Size

Abstract: In this article, we propose tests for covariance matrices of high dimension with fewer observations than the dimension for a general class of distributions with positive definite covariance matrices. In one-sample case, tests are proposed for sphericity and for testing the hypothesis that the covariance matrix ∑ is an identity matrix, by providing an unbiased estimator of tr [∑ 2 ] under the general model which requires no more computing time than the one available in the literature for normal model. In the two-sample case, tests for the equality of two covariance matrices are given. The asymptotic distributions of proposed tests in one-sample case are derived under the assumption that the sample size N = O ( p δ ), 1/2 < δ < 1, where p is the dimension of the random vector, and O ( p δ ) means that N/p goes to zero as N and p go to infinity. Similar assumptions are made in the two-sample case.

Theory and simulations of covariance mapping in multiple dimensions for data analysis in high-event-rate experiments

Abstract: Multidimensional covariance analysis and its validity for correlation of processes leading to multiple products are investigated from a theoretical point of view. The need to correct for false correlations induced by experimental parameters which fluctuate from shot to shot, such as the intensity of self-amplified spontaneous emission x-ray free-electron laser pulses, is emphasized. Threefold covariance analysis based on simple extension of the two-variable formulation is shown to be valid for variables exhibiting Poisson statistics. In this case, false correlations arising from fluctuations in an unstable experimental parameter that scale linearly with signals can be eliminated by threefold partial covariance analysis, as defined here. Fourfold covariance based on the same simple extension is found to be invalid in general. Where fluctuations in an unstable parameter induce nonlinear signal variations, a technique of contingent covariance analysis is proposed here to suppress false correlations. In this paper we also show a method to eliminate false correlations associated with fluctuations of several unstable experimental parameters.

A semiparametric approach to simultaneous covariance estimation for bivariate sparse longitudinal data.

Abstract: Estimation of the covariance structure for irregular sparse longitudinal data has been studied by many authors in recent years but typically using fully parametric specifications. In addition, when data are collected from several groups over time, it is known that assuming the same or completely different covariance matrices over groups can lead to loss of efficiency and/or bias. Nonparametric approaches have been proposed for estimating the covariance matrix for regular univariate longitudinal data by sharing information across the groups under study. For the irregular case, with longitudinal measurements that are bivariate or multivariate, modeling becomes more difficult. In this article, to model bivariate sparse longitudinal data from several groups, we propose a flexible covariance structure via a novel matrix stick-breaking process for the residual covariance structure and a Dirichlet process mixture of normals for the random effects. Simulation studies are performed to investigate the effectiveness of the proposed approach over more traditional approaches. We also analyze a subset of Framingham Heart Study data to examine how the blood pressure trajectories and covariance structures differ for the patients from different BMI groups (high, medium, and low) at baseline.

Properties of spectral covariance for linear processes with infinite variance

Abstract: We consider a measure of dependence for symmetric α-stable random vectors, which was introduced by the second author in 1976. We demonstrate that this measure of dependence, which we suggest to call the spectral covariance, can be extended to random vectors in the domain of normal attraction of general stable vectors. We investigate the asymptotic of the spectral covariance function for linear stable (Ornstein–Uhlenbeck, log-fractional, linear-fractional) processes with infinite variance and show that, in comparison with the results on the properties of codifference of these processes, obtained two decades ago, the results for the spectral variance are obtained under more general conditions and calculations are simpler.

Radial Covariance Functions Motivated by Spatial Random Field Models with Local Interactions

Abstract: We derive explicit expressions for a family of radially symmetric, non-differentiable, Spartan covariance functions in $\mathbb{R}^2$ that involve the modified Bessel function of the second kind. In addition to the characteristic length and the amplitude coefficient, the Spartan covariance parameters include the rigidity coefficient $\eta_{1}$ which determines the shape of the covariance function. If $ \eta_{1} >> 1$ Spartan covariance functions exhibit multiscaling. We also derive a family of radially symmetric, infinitely differentiable Bessel-Lommel covariance functions valid in $\mathbb{R}^{d}, d\ge 2$. We investigate the parametric dependence of the integral range for Spartan and Bessel-Lommel covariance functions using explicit relations and numerical simulations. Finally, we define a generalized spectrum of correlation scales $\lambda^{(\alpha)}_{c}$ in terms of the fractional Laplacian of the covariance function; for $0 \le \alpha \le1$ the $\lambda^{(\alpha)}_{c}$ extend from the smoothness microscale $(\alpha=1)$ to the integral range $(\alpha=0)$. The smoothness scale of mean-square continuous but non-differentiable random fields vanishes; such fields, however, can be discriminated by means of $\lambda^{(\alpha)}_{c}$ scales obtained for $\alpha <1$.

Efficient recovery of principal components from compressive measurements with application to Gaussian mixture model estimation

Abstract: There has been growing interest in performing signal processing tasks directly on compressive measurements, e.g. low-dimensional linear measurements of signals taken with Gaussian random vectors. In this paper, we present a highly efficient algorithm to recover the covariance matrix of high-dimensional data from compressive measurements. We show that, as the number of data samples increases, the eigenvectors (principal components) of the empirical covariance matrix of a simple matrix-vector multiplication of the compressive measurements converge to the true principal components of the original data. Also, we investigate the perturbation of eigenvalues of the covariance matrix under random projection of the data to find conditions for approximate recovery of them. Furthermore, we introduce an important application of our proposed method for efficient estimation of the parameters of Gaussian Mixture Models from compressive measurements. We present experimental results demonstrating the performance and efficiency of our proposed algorithms.

On estimation of mean and covariance functions in repeated time series with long-memory errors*

Abstract: We consider kernel estimation of trend and covariance functions in models typically encountered in functional data analysis (FDA), with the modification that the random curves are perturbed by error processes that exhibit short- or long-range dependence. Uniform convergence of standardized maximal differences between estimated and true (trend and covariance) functions is established. For the covariance function, a transformation based on contrasts is proposed that does not require explicit trend estimation. Improved estimators can be obtained by using higher-order kernels.

Optimal estimation and filtration under unknown covariances of random factors

Abstract: The general schemes of linear estimation and filtration were considered on assumption of the unknown covariance matrix of random factors such as unknown parameters, measurement errors, and initial and external perturbations. A new criterion was introduced for the quality of estimate or filter. It is the level of damping random perturbations which is defined by the maximal value over all covariance matrices of the root-mean-square error normalized by the sum of variances of all random factors. The level of damping random perturbations was shown to be equal to the square of the spectral norm of the matrix relating the error of estimation and the random factors, and the optimal estimate minimizing this criterion was established. In the problem of filtration, it was shown how the filter parameters that are optimal in the level of damping random perturbations are expressed in terms of the linear matrix inequalities.

Covariance Matrix Functions of Isotropic Vector Random Fields

Abstract: An isotropic scalar or vector random field is a second-order random field in (d ⩾ 2), whose covariance function or direct/cross covariance functions are isotropic While isotropic scalar random fields have been well developed and widely used in various sciences and industries, the theory of isotropic vector random fields needs to be investigated for applications The objective of this article is to study properties of covariance matrix functions associated with vector random fields in which are stationary, isotropic, and mean square continuous, and derives the characterizations of the covariance matrix function of the Gaussian or second-order elliptically contoured vector random field in In particular, integral or spectral representations for isotropic and continuous covariance matrix functions are derived

Fluctuations for linear spectral statistics of large random covariance matrices

Abstract: The theory of large random matrices has proved to be an efficient tool to address many problems in wireless communication and statistical signal processing these last two decades. We provide hereafter a central limit theorem (CLT) for linear spectral statistics of large random covariance matrices, improving Bai and Silverstein's celebrated 2004 result. This fluctuation result should be of interest to study the fluctuations of important estimators in statistical signal processing.

Modeling spatial covariance functions

Abstract: Choi, InKyung Ph.D., Purdue University, December 2014. Modeling spatial covariance functions. Major Professor: Hao Zhang. Covariance modeling plays a key role in the spatial data analysis as it provides important information about the dependence structure of underlying processes and determines performance of spatial prediction. Various parametric models have been developed to accommodate the idiosyncratic features of a given dataset. However, the parametric models may impose unjustified restrictions to the covariance structure and the procedure of choosing a specific model is often ad-hoc. In the first part of the dissertation, a new nonparametric covariance model that can avoid the choice of parametric forms is proposed. The estimator is obtained via a nonparametric approximation of completely monotone functions. It is easy to implement and simulation study shows it outperforms the parametric models when there is no clear information on model specification. Two real datasets are analyzed to illustrate the proposed approach and provide further comparison between the nonparametric and parametric models. Most spatial covariance models assume that the dependence becomes stronger when two locations are closer to each other and thus assume that the dependence is negligible when two locations are far apart from one another. However long-distance connection can occur in climate variables through, for example, high altitude winds or large-scale atmospheric waves propagation. This phenomenon is called teleconnection and often considered to be responsible for extreme weather events occurring simultaneously around the world. In the second part of the dissertation, a nonstationary spatial covariance model for long-distance dependence is proposed. The model allows the spatial dependence to vary with time so that temporal dynamics of the telecon-

Sensitivity analysis for the identifiability with application to latent random effect model for the mixed data

Abstract: In this paper, we study the indentifiability of a latent random effect model for the mixed correlated continuous and ordinal longitudinal responses. We derive conditions for the identifiability of the covariance parameters of the responses. Also, we proposed sensitivity analysis to investigate the perturbation from the non-identifiability of the covariance parameters, it is shown how one can use some elements of covariance structure. These elements associate conditions for identifiability of the covariance parameters of the responses. Influence of small perturbation of these elements on maximal normal curvature is also studied. The model is illustrated using medical data.

Regularized robust estimation of mean and covariance matrix under heavy tails and outliers

Abstract: In this paper we consider the regularized mean and covariance estimation problem for samples drawn from elliptical family of distributions. The proposed estimator yields robust estimates when the underlying distribution is heavy-tailed or when there are outliers in the data samples. In the scenario that the number of samples is small, it shrinks the estimator of the mean and covariance towards arbitrary given prior targets. Numerical algorithms are designed for the estimator based on the majorization-minimization framework and the simulation shows that the proposed estimator achieves considerably better performance.

Covariate‐based cepstral parameterizations for time‐varying spatial error covariances

Abstract: The difference between a mechanistic model of a spatio-temporal process and associated observations can reasonably be represented by a random process with possible dependence structure in space and time. Often, such error processes are assumed to be stationary in time, so that a unique spatial error covariance structure is applicable for all time. However, model performance can be related to large-scale time-varying features, causing the error process to be nonstationary, leading to a time-varying spatial covariance. Because of dimensionality constraints, one cannot estimate such time-varying covariances reliably, and therefore they must be modeled. Such models for covariances should be able to accommodate influences from exogenous covariates. However, the inclusion of covariates in a standard covariance model is challenging due to the fact that the covariance must be positive definite. We mitigate this issue by modeling the time-varying spatial dependence through the Fourier coefficients of the log spectrum, as originally developed for time series representations. The convenient feature of these so-called “cepstral” models (or “exponential models”) is that the associated parameters are unrestricted yet still are guaranteed to produce a positive definite covariance. Thus, it is fairly straightforward to model these parameters as time-varying random processes that depend on other covariates within the Bayesian hierarchical modeling paradigm. We illustrate the effectiveness of this model through various simulated examples and by applying it to differences from a long-lead sea surface temperature forecast model. In this case, we consider the Pacific Decadal Oscillation and Southern Oscillation Index as possible factors that influence the error covariance. Copyright © 2014 John Wiley & Sons, Ltd.

The derivation of mutual information and covariance function using centered random variables

Abstract: Information theoretic measures such as Mutual Information are often said to be able to measure nonlinear dependencies whereas covariance (and correlation) are able to measure only linear dependencies. We aim to illustrate this claim using centered random variables. The set of centered random variable Fc = {−q−12,−q−12+1,...,q−12−1,q−12} is mapped from F = {1,2, ..., q − 1, q}. For q=2, we derive the relationship between the Mutual Information function, I, and the covariance function, Γ, and show that Γ=0→I=0. Furthermore we show that when q=3, the nonlinearities are captured by Mutual Information by highlighting a case where Γ=0 ⇸ I=0.

An Optimal Estimate for the Covariance of Indicator Functions of Associated Random Variables

Abstract: An optimal estimate for the covariance of indicator functions of positively or negatively associated random variables which improves an inequality of Bagai and Prakasa Rao is obtained. Applications of the proved result to investigation of the limit properties of empirical distribution functions and volumes of excursion sets of random fields are considered.

A Generalized Convolution Model and Estimation for Non-stationary Random Functions

Abstract: Standard geostatistical models assume second order stationarity of the underlying Random Function. In some instances, there is little reason to expect the spatial dependence structure to be stationary over the whole region of interest. In this paper, we introduce a new model for second order non-stationary Random Functions as a convolution of an orthogonal random measure with a spatially varying random weighting function. This new model is a generalization of the common convolution model where a non-random weighting function is used. The resulting class of non-stationary covariance functions is very general, flexible and allows to retrieve classes of closed-form non-stationary covariance functions known from the literature, for a suitable choices of the random weighting functions family. Under the framework of a single realization and local stationarity, we develop parameter inference procedure of these explicit classes of non-stationary covariance functions. From a local variogram non-parametric kernel estimator, a weighted local least-squares approach in combination with kernel smoothing method is developed to estimate the parameters. Performances are assessed on two real datasets: soil and rainfall data. It is shown in particular that the proposed approach outperforms the stationary one, according to several criteria. Beyond the spatial predictions, we also show how conditional simulations can be carried out in this non-stationary framework.

Mapping the decay of double core hole states of atoms and molecules

Abstract: Covariance mapping is used to investigate electron emission of double core hole states and associated decay processes of atoms and molecules upon absorption of multiple X-ray photons provided by an FEL source.

Models of covariance functions of gaussian random fields escaping from isotropy, stationarity and non negativity

Abstract: This paper represents a survey of recent advances in modeling of space or space-time Gaussian Random Fields (GRF), tools of Geostatistics at hand for the understanding of special cases of noise in image analysis. They can be used when stationarity or isotropy are unrealistic assumptions, or even when negative covariance between some couples of locations are evident. We show some strategies in order to escape from these restrictions, on the basis of rich classes of well known stationary or isotropic non negative covariance models, and through suitable operations, like linear combinations, generalized means, or with particular Fourier transforms.

Analysis of Eigenspace Dynamics with Applications to Array Processing

Abstract: : For an N-element array (Fig.1(a)), methods such as beamforming and singular value decomposition rely on estimation of the sample covariance matrix, computed from M independent data snapshots. As M , the sample covariance is a consistent estimator of the true population covariance. However, this ideal condition cannot be met in most practical situations,1-2 in which large-aperture arrays operate in the presence of fast maneuvering interferers, or with towed/drifting arrays strongly affected by deformation or array-depth perturbations. The long-term goal of this effort is the development of physically motivated models to statistically describe the eigenstructure (eigenvalues and eigenvectors) of sample covariance matrices in sample-starved settings, and the use of those models for performance analysis and improvement of array processing methods. To this end, mathematical tools developed in the context of Random Matrix Theory (RMT)3-6 (mostly focused in the regime NM) and High Dimension, Low Sample Size (HDLSS) array processing7-8 (which considers NM) are applied to obtain statistical descriptions of sample eigenvalues/eigenvectors and how those quantities differ from the (true) population eigenpairs. Additional long-term goals are exploiting the information carried by sample eigenvectors for the improvement of estimators of the sample covariance matrix (i.e., signal versus noise subspaces), and for quantifying local stationary in array data (Fig.1 (b)).

Utilizing wind in spatial covariance

Abstract: This work develops a covariance function which allows for a stronger spatial correlation for pairs of points in the direction of a vector such as wind and weaker for pairs which are perpendicular to it. It derives a simple covariance function by stretching the space along the wind axes (upwind and across wind axes). It is shown that this covariance function is anisotropy in the original space and the functions is explicitly calculated.